Abstract
AbstractIn this paper, we consider continuous‐time Markov chains with a finite state space under nonlinear expectations. We define so‐called Q‐operators as an extension of Q‐matrices or rate matrices to a nonlinear setup, where the nonlinearity is due to model uncertainty. The main result gives a full characterization of convex Q‐operators in terms of a positive maximum principle, a dual representation by means of Q‐matrices, time‐homogeneous Markov chains under convex expectations, and a class of nonlinear ordinary differential equations. This extends a classical characterization of generators of Markov chains to the case of model uncertainty in the generator. We further derive an explicit primal and dual representation of convex semigroups arising from Markov chains under convex expectations via the Fenchel–Legendre transformation of the generator. We illustrate the results with several numerical examples, where we compute price bounds for European contingent claims under model uncertainty in terms of the rate matrix.
Highlights
AND MAIN RESULTIn mathematical finance, model uncertainty or ambiguity is an almost omnipresent phenomenon, which, for example, appears due to incomplete information about certain aspects of an underlying asset or insufficient data in order to perform reliable statistical estimation methods for the parameters of a stochastic process
While these works give sufficient conditions in order to guarantee the existence of stochastic processes under model uncertainty and to establish a connection to nonlinear partial differential equations, there is no necessary condition that determines the maximal degree of ambiguity that can be captured by an uncertain process
In particular, computational aspects of sublinear imprecise Markov chains have been studied amongst others by Krak, De Bock, and Siebes (2017) and Škulj (2015). Another concept that is closely related to Markov chains under nonlinear expectations, as discussed in the present paper, are backward stochastic differential equations (BSDEs) on Markov chains by Cohen and Elliott (2008) and Cohen and Elliott (2010a), see Cohen and Szpruch (2012), Cohen and Hu (2013), and Cohen and Elliott (2010b) for the discrete-time case
Summary
Model uncertainty or ambiguity is an almost omnipresent phenomenon, which, for example, appears due to incomplete information about certain aspects of an underlying asset or insufficient data in order to perform reliable statistical estimation methods for the parameters of a stochastic process. In particular, computational aspects of sublinear imprecise Markov chains have been studied amongst others by Krak, De Bock, and Siebes (2017) and Škulj (2015) Another concept that is closely related to Markov chains under nonlinear expectations, as discussed in the present paper, are BSDEs on Markov chains by Cohen and Elliott (2008) and Cohen and Elliott (2010a), see Cohen and Szpruch (2012), Cohen and Hu (2013), and Cohen and Elliott (2010b) for the discrete-time case. We derive an explicit dual representation in terms of an optimal control problem, where nature tries to control the system into the worst possible scenario, giving a control-theoretic interpretation to Markov chains under convex expectations
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
